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Mirrors > Home > MPE Home > Th. List > cjsub | Structured version Visualization version Unicode version |
Description: Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.) |
Ref | Expression |
---|---|
cjsub |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negcl 10281 | . . 3 | |
2 | cjadd 13881 | . . 3 | |
3 | 1, 2 | sylan2 491 | . 2 |
4 | negsub 10329 | . . 3 | |
5 | 4 | fveq2d 6195 | . 2 |
6 | cjneg 13887 | . . . . 5 | |
7 | 6 | adantl 482 | . . . 4 |
8 | 7 | oveq2d 6666 | . . 3 |
9 | cjcl 13845 | . . . 4 | |
10 | cjcl 13845 | . . . 4 | |
11 | negsub 10329 | . . . 4 | |
12 | 9, 10, 11 | syl2an 494 | . . 3 |
13 | 8, 12 | eqtrd 2656 | . 2 |
14 | 3, 5, 13 | 3eqtr3d 2664 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cfv 5888 (class class class)co 6650 cc 9934 caddc 9939 cmin 10266 cneg 10267 ccj 13836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-2 11079 df-cj 13839 df-re 13840 df-im 13841 |
This theorem is referenced by: sqabssub 14023 cjcn2 14330 mul4sqlem 15657 dvcjbr 23712 isosctrlem2 24549 atancj 24637 dipsubdi 27704 his2sub2 27950 sigarmf 41043 |
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